Complexity and cohomology for cut and projection tilings

TL;DR

This paper investigates the complexity and cohomology of cut and projection tilings, establishing polynomial growth of complexity, computing its exponent, and exploring the relationship with cohomology properties.

Contribution

It provides a detailed analysis of the complexity function for cut and projection tilings, including explicit calculations and bounds for the growth exponent, and links this to cohomological properties.

Findings

Complexity function exhibits polynomial growth with computable exponent.

The exponent relates to the ranks of certain algebraic groups in the tiling construction.

A counter-example shows the link between complexity and cohomology does not hold for all tilings.

Abstract

We consider a subclass of tilings, the tilings obtained by cut and projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent \alpha in terms of the ranks of certain groups which appear in the construction. We give bounds for \alpha. These computations apply to some well known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over \Q. We show that such a link cannot be established for more general tilings, and we present a counter-example in dimension one.